1. **Stating the problem:** We are asked to find the characteristic equations of given 2x2 matrices. 2. **Formula:** The characteristic equation of a matrix $A$ is given by $$\det(A - \lambda I) = 0,$$ where $\lambda$ is an eigenvalue and $I$ is the identity matrix. 3. **Step-by-step for matrix (a):** Matrix (a): $$\begin{bmatrix}3 & 0 \\ 8 & -1\end{bmatrix}$$ Calculate $$\det\left(\begin{bmatrix}3-\lambda & 0 \\ 8 & -1-\lambda\end{bmatrix}\right) = (3-\lambda)(-1-\lambda) - 0 \cdot 8 = 0.$$ Expand: $$(3-\lambda)(-1-\lambda) = 0$$ $$-3 - 3\lambda + \lambda + \lambda^2 = 0$$ $$\lambda^2 - 2\lambda - 3 = 0$$ 4. **Matrix (b):** $$\begin{bmatrix}10 & -9 \\ 4 & -2\end{bmatrix}$$ Characteristic polynomial: $$\det\left(\begin{bmatrix}10-\lambda & -9 \\ 4 & -2-\lambda\end{bmatrix}\right) = (10-\lambda)(-2-\lambda) - (-9)(4) = 0$$ Expand: $$(10-\lambda)(-2-\lambda) + 36 = 0$$ $$-20 - 10\lambda + 2\lambda + \lambda^2 + 36 = 0$$ $$\lambda^2 - 8\lambda + 16 = 0$$ 5. **Matrix (c):** $$\begin{bmatrix}0 & 3 \\ 4 & 0\end{bmatrix}$$ Characteristic polynomial: $$\det\left(\begin{bmatrix}-\lambda & 3 \\ 4 & -\lambda\end{bmatrix}\right) = (-\lambda)(-\lambda) - 3 \cdot 4 = 0$$ $$\lambda^2 - 12 = 0$$ 6. **Matrix (d):** $$\begin{bmatrix}-2 & -7 \\ 1 & 2\end{bmatrix}$$ Characteristic polynomial: $$\det\left(\begin{bmatrix}-2-\lambda & -7 \\ 1 & 2-\lambda\end{bmatrix}\right) = (-2-\lambda)(2-\lambda) - (-7)(1) = 0$$ Expand: $$(-2-\lambda)(2-\lambda) + 7 = 0$$ $$-4 + 2\lambda - 2\lambda + \lambda^2 + 7 = 0$$ $$\lambda^2 + 3 = 0$$ 7. **Matrix (e):** $$\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$$ Characteristic polynomial: $$\det\left(\begin{bmatrix}-\lambda & 0 \\ 0 & -\lambda\end{bmatrix}\right) = (-\lambda)(-\lambda) - 0 = \lambda^2 = 0$$ 8. **Matrix (f):** $$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$ Characteristic polynomial: $$\det\left(\begin{bmatrix}1-\lambda & 0 \\ 0 & 1-\lambda\end{bmatrix}\right) = (1-\lambda)^2 = 0$$ **Final characteristic equations:** (a) $$\lambda^2 - 2\lambda - 3 = 0$$ (b) $$\lambda^2 - 8\lambda + 16 = 0$$ (c) $$\lambda^2 - 12 = 0$$ (d) $$\lambda^2 + 3 = 0$$ (e) $$\lambda^2 = 0$$ (f) $$(1-\lambda)^2 = 0$$